[1]} introduced Canonical Correlation Analysis (CCA) as a dimensionality reduction method that projects two sets of random variables (i.e. two data sets) into a shared subspace such that the projections are maximally correlated. Although almost always commonly thought as data reduction technique, CCA can indeed be given a latent variable interpretation as [2]} have shown. More specifically, they showed that the equivalent probabilistic latent variable model can be formulated as
\(\begin{split}\mathbf {z}_n&\sim \mathcal {N}_d\left(0,\mathbf {I}\right) \quad n=1, \cdots ,N\\\begin{bmatrix}{\mathbf {y}_n}^{(1)}\mid \mathbf {z}_n\\{\mathbf {y}_n}^{(2)}\mid \mathbf {z}_n\end{bmatrix} &\sim \mathcal {N}_{(p1+p2)}\left(\begin{bmatrix}\mathbf {W}^{(1)}\\\mathbf {W}^{(2)}\end{bmatrix} \mathbf {z}_n,\begin{bmatrix}\Sigma ^{(1)} & 0\\0 & \Sigma ^{(2)}\end{bmatrix}\right)\end{split}\) 
